Origin of name is Serbian, not Russian or Bulgarian. They merely adopted this name. It is translation from Greek name Theodosios and it means (as well as Theodosios) "Given by God". Bogdan = "Bog" (God) + "dan" (given) and not "God's gift" or "Giving God" as you translated Theodosios. Theodore would be the "Gift of God" and we have direct translation of this name in Serbian and it is Božidar. Božiji(God's) + dar (gift)= Božidar. So both names are Serbian origin.
Name variant: Bogden. During WWII, basic supplies were rationed and people often ran short. Feuds on occassions erupted between the haves and the have nots. In one case, this family name separated ways distinguishing the difference by changing the "A" to an "E".